Abstract
We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.
Highlights
The aim of this paper is to show that the play and the stop operator possess Newton as well as Bouligand derivatives, and to compute those derivatives
Bouligand derivatives are closely related to Newton derivatives, and can be used to provide sensitivity results as well as optimality conditions for problems involving nonsmooth operators
Their formal definition, in the spirit of [11, 12], is given below in Section 6; alternatively, they arise as solution operators of the evolution variational inequality w (t) · (ζ − z(t)) ≤ 0, for all ζ ∈ [−r, r], z(t) ∈ [−r, r], z(a) = z0 ∈ [−r, r], w(t) + z(t) = u(t)
Summary
The aim of this paper is to show that the play and the stop operator possess Newton as well as Bouligand derivatives, and to compute those derivatives. The number z0 plays the role of an initial condition Their formal definition, in the spirit of [11, 12], is given below in Section 6; alternatively, they arise as solution operators of the evolution variational inequality w (t) · (ζ − z(t)) ≤ 0 , for all ζ ∈ [−r, r], z(t) ∈ [−r, r] , z(a) = z0 ∈ [−r, r] , w(t) + z(t) = u(t). The results below serve to narrow the gap between differentiability and non-differentiability of rateindependent operators Their proofs given here are based on the same idea as used in [3], namely, to locally represent the play as a composition of operators whose main ingredient is the cumulated maximum. Our main results are given in Theorem 7.15 for Newton differentiability and Theorem 8.2 for Bouligand differentiability of the play They are based on corresponding results for the maximum functional We have chosen to elaborate the proofs for both cases to some extent; the details are somewhat cumbersome and should not be placed too much as a burden on the reader
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